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Chapter 5: Problem 39

The sum of the squares of two positive numbers is 29 and the difference of the squares of the numbers is 21 . Find the numbers.

Short Answer

Expert verified
The numbers are 5 and 2.

Step by step solution

01

- Set up the equations

Let the two positive numbers be denoted by x and y. According to the problem, the sum of the squares of the two numbers is 29, which gives us our first equation: x^2 + y^2 = 29. The problem also states that the difference of the squares of the two numbers is 21, resulting in the second equation: x^2 - y^2 = 21.
02

- Solve for one variable

Add the two equations to eliminate y^2: (x^2 + y^2) + (x^2 - y^2) = 29 + 21. This simplifies to 2x^2 = 50, so x^2 = 25. Taking the square root of both sides, we get x = 5.
03

- Substitute back to find the second variable

Use x = 5 in one of the original equations. Substituting into x^2 - y^2 = 21, we get 25 - y^2 = 21. Simplifying, we find y^2 = 4, and taking the square root of both sides, y = 2.
04

- Verify the solution

Check the values of x and y in the other original equation to ensure they satisfy both equations. Substituting x = 5 and y = 2 into x^2 + y^2 = 29, we get 25 + 4 = 29, which is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Equations
Algebraic equations are mathematical statements that show the equality between two expressions. In the given exercise, we deal with quadratic equations, which are equations involving squares of variables. Setting up algebraic equations correctly is essential for solving problems. Here, we have two equations: one for the sum of squares and one for the difference of squares.
By declaring the variables: \( x^2 + y^2 = 29 \) and \( x^2 - y^2 = 21 \), we create a system of equations that can be solved step by step.
This setup is crucial to finding a logical path to the solution.
Sum of Squares
The 'sum of squares' refers to an equation where we add the squares of numbers. In this problem, we know the sum of squares of two positive numbers equals 29.
This means if \ x \ and \ y \ are our numbers, \( x^2 + y^2 = 29 \). Summing squares is common in algebra and is useful in various applications like finding distances in geometry.
To understand how sum of squares works, consider adding individual squares of numbers and examining their properties. In our problem, it helps set up one of the key equations needed to solve for \ x \ and \ y \.
Difference of Squares
The 'difference of squares' is a unique algebraic structure that describes the difference between squares of two numbers. It has a special factorization: \ (a^2 - b^2 = (a-b)(a+b)) \.
In our problem, the difference of squares of two numbers is given as 21, meaning \( x^2 - y^2 = 21 \). This can help us isolate one of the variables.
By adding and subtracting our equations wisely, we can eliminate variables step by step, simplifying the problem. The difference of squares concept allows breaking down complex problems into simpler parts.
Problem-Solving Techniques
Problem-solving in algebra involves various techniques such as elimination, substitution, and verification. Here are some key strategies used in the exercise:
  • Setting Up Equations: Define variables and use given conditions to form equations.
  • Elimination: Add or subtract equations to eliminate one variable, simplifying the system.
  • Substitution: Once a variable is isolated, substitute its value back into one of the original equations to find the other variable.
  • Verification: Always check the solution in both original equations to ensure correctness.
By following these methods, we methodically approach and solve algebraic problems.鈥� } ] }

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Most popular questions from this chapter

Explain how test points are used to determine the region of the plane that represents the solution to an inequality in two variables.

Determine if the ordered pair is a solution to the system of equations. (See Example 1\()\) \(-11 x+6 y=-4\) \(7 x+3 y=23\) a. \(\left(1, \frac{7}{6}\right)\) b. (2,3)

Solve the system. $$ \begin{array}{l} 2^{x}+2^{y}=6 \\ 4^{x}-2^{y}=14 \end{array} $$

Use substitution to solve the system for the set of ordered triples \((x, y, \lambda)\) that satisfy the system. $$ \begin{array}{l} 8=4 \lambda x \\ 2=2 \lambda y \\ 2 x^{2}+y^{2}=9 \end{array} $$

A manufacturer produces two models of a gas grill. Grill A requires 1 hr for assembly and \(0.4 \mathrm{hr}\) for packaging. Grill \(B\) requires 1.2 hr for assembly and 0.6 hr for packaging. The production information and profit for each grill are given in the table. (See Example 4\()\) $$ \begin{array}{|l|c|c|c|} \hline & \text { Assembly } & \text { Packaging } & \text { Profit } \\ \hline \text { Grill A } & 1 \mathrm{hr} & 0.4 \mathrm{hr} & \$ 90 \\ \hline \text { Grill B } & 1.2 \mathrm{hr} & 0.6 \mathrm{hr} & \$ 120 \\ \hline \end{array} $$ The manufacturer has \(1200 \mathrm{hr}\) of labor available for assembly and \(540 \mathrm{hr}\) of labor available for packaging. a. Determine the number of grill A units and the number of grill B units that should be produced to maximize profit assuming that all grills will be sold. b. What is the maximum profit under these constraints? c. If the profit on grill A units is $$\$ 110$$ and the profit on grill \(\underline{B}\) units is unchanged, how many of each type of grill unit should the manufacturer produce to maximize profit?
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